Function uw12::four_el::form_fock_four_el_df

Function Documentation

utils::FockMatrixAndEnergy uw12::four_el::form_fock_four_el_df(const integrals::Integrals &W, const integrals::Integrals &V, bool indirect_term, bool calculate_fock, double scale_opp_spin, double scale_same_spin)

Find the contribution to the energy and Fock matrix of the four electron term of UW12.

Calculates the four electron UW12 Fock matrix and energy using density fitting integrals.

In this method, the direct (+) and indirect (-) contributions are calculated separately with the energies given by

\[\begin{align*}E_{c, 4el,+}^{UW12} &= \frac{1}{2} \sum_{ijkl} \left( ik \left| w_{12}^{s_{ij}} \right| jl \right) \left( ki \left| r_{12}^{-1} \right| lj \right) \newline &= \frac{1}{2} \sum_{Cik} \left( ik \left| w_{12}^{s_{ij}} \right| C \right) \sum_{D} \left( ki \left| r_{12}^{-1} \right| D \right) \sum_{jl} \left( \tilde{D} \left| r_{12}^{-1} \right| lj \right) \left( \tilde{C} \left| w_{12}^{s_{ij}} \right| jl \right) \newline E_{c, 4 el, -}^{UW12} &= \frac{1}{2} \sum_{ijkl} \left( il \left| w_{12}^{1} \right| jk \right) \left( ki \left| r_{12}^{-1} \right| lj \right), \end{align*}\]
where the indirect integrals must be calculated separately using density fitting and
\[\left( \tilde{C} \left| x \right| i j \right) = \sum_{D} \left( C \left| X \right\vert D \right)^{-1} \left(D \left| X \right| i j \right) \]
for potential \(X\). For the frozen-core approximation, orbital indices i and j run over active (valence) orbitals only, while k and l include the core orbitals in the summation.

The corresponding Fock contributions are given by

\[\begin{align*}\frac{\partial E_{c, 4 el, +}^{UW12}}{\partial D_{\alpha \beta}^{\sigma}} &= \sum_{C k} \left( \alpha k \left| w_{12}^{s_{ij}} \right| C \right) \sum_{D} \left( k\beta \left| r_{12}^{-1} \right| D \right) \sum_{jl} \left( \tilde{D} \left| r_{12}^{-1} \right| lj \right) \left( \tilde{C} \left| w_{12}^{s_{ij}} \right| jl \right) \newline &\qquad + \sum_{C i} \left( i \beta \left| w_{12}^{s_{ij}} \right| C \right) \sum_{D} \left( \alpha i \left| r_{12}^{-1} \right| D \right) \sum_{jl} \left( \tilde{D} \left| r_{12}^{-1} \right| lj \right) \left( \tilde{C} \left| w_{12}^{s_{ij}} \right| jl \right) \newline \frac{\partial E_{c, 4 el,-}^{UW12}}{\partial D_{\alpha \beta}^{\sigma}} &= - \sum_{jkl} \delta_{\sigma_{j} \sigma} \left( \alpha l \left| w_{12}^{1} \right| jk \right) \left( k \beta \left| r_{12}^{-1} \right| l j\right) - \sum_{ijk} \delta_{\sigma_{j} \sigma} \left( \alpha i \left| w_{12}^{1} \right| kj \right) \left( j \beta \left| r_{12}^{-1} \right| ik \right) \end{align*}\]
where the indirect four component integral must be calculated separately. In the full-core Unsold-W12, both terms in each of these equations are equal. However, for the frozen-core approximation they are not, since the summations are now over different orbitals.

Since the four component integrals must be constructed separately for the indirect term, this algorithm scales as \(N^5\). However, the indirect term is for same spin only. Therefore, for \(w^{s=1}(r) = 0\) (opposite spin only), this term is zero, and the algorithm is \(N^4\).

Parameters

  • W – \(w(r)\) integrals

  • V – \(r^{-1}\) integrals

  • indirect_term – Whether to calculate the indirect term

  • calculate_fock – Whether to calculate the Fock matrix

  • scale_opp_spin – Scale for opposite spin

  • scale_same_spin – Scale for same spin

Returns

Four electron UW12 Fock matrix and energy